Atom matrix

Intra-atomic matrix elements, or on-site tenns, with / = m.. Traditionally, the potential contributions... 

Mercer J L Jr and Chou M Y 1994 Tight-binding model with intra-atomic matrix elements Phys. Rev. B 49 8506... 

Metal atom matrix chemistry. Correlation of bonding with chemisorbed molecules. G. A. Ozin, Acc. Chem. Res., 1977,10, 21-26 (32). 

By use of quantitative, metal-atom, matrix-cocondensation techniques and the kinetic analysis previously discussed, the dimeric spe-... 

Techniques other than UV-visible spectroscopy have been used in matrix-isolation studies of Ag see, for example, some early ESR studies by Kasai and McLeod 56). The fluorescence spectra of Ag atoms isolated in noble-gas matrices have been recorded (76,147), and found to show large Stokes shifts when optically excited via a Si j — atomic transition which is threefold split in the matrix by spin-orbit and vibronic interactions. The large Stokes shifts may be explained in terms of an excited state silver atom-matrix cage complex in this... 

Fig. 11. Photograph of the four-electrode, vacuum flange and dual, quartz crystal, microbalance assembly, (A) side view, and (B) front view, used for mixed Cr atom. Mo atom matrix depositions with simultaneous monitoring of the individual metal flows. (The resolution of the microbalance is 10 g) (113).

A wide range of ethylene complexes, both mononuclear and higher cluster in nature, have been synthesized, and studied, by the metal atom-matrix technique. In this Section, we shall focus on the reactions... 

Nonmetal-atom, matrix-isolation spectroscopy has proved useful in structure and isomer determination of stable, metal carbonyls. Fe(CO)4(NO) was investigated (157) in low-temperature matrices with CO enrichment, and it was demonstrated that the IR spectrum is consistent with C v symmetry (trigonal bipyramid with an equatorial NO), in agreement with X-ray studies (55). The work resolves the dis-... 

A number of other metal atom-matrix studies have appeared in the literature, with such typical inorganic ligands as NO and H. In the following Section, we shall briefly summarize some of these results. 

The concept of a closed system can also be introduced without considering reactions. Chemical species are built from building blocks called atoms. Define the atom matrix A, where [A] j is the number of the i-th atom in the molecule of the j-th species Mj. If the number of different atoms is denoted by a then the atom matrix is of dimension a k. The quantities of atoms in the system can be calculated by summing up their quantities in each species, i.e., forming the product An. These quantities remain unchanged if the system is closed, so that... 

Equation (1.76) expresses the fundamental relation between the atom matrix and the reaction matrix of a closed system. The matrices A and , however, result in the same stoichiometric subspace if and only if the subspace defined by (1.73) and the one defined by (1.74) are of the same dimension, in addition to the relation (1.76). I4e denote the dimension of the stoichiometric subspace by f also called the stoichiometric number of freedom. If the reaction matrix is known, then f = rank(B),... 

Notice that the number NA of atoms is usually small compared to the number NS of species, and hence the RAND algorithm is very effective in terms of computational effort. The rank of the atom matrix, however, must be equal to the number NA of atoms. At this point it is interesting to remark that instead of the atom matrix we can use a virtual atom matrix, i.e., the matrix of reaction invariant coefficients if the atom matrix is not available or we are interested in a restricted equilibrium. For details see Section 1.8.1. 

The input is accepted in chemical notation. The atom matrix is constructed in the "formula interpreter" section in lines 214-248. Strictly speaking the function we minimize is not defined if any one of the mole numbers is zero. Since the vector of initial mole numbers serves also as an initial guess of the solution and it often contains zero elements, we add a small quantity to each... 

In the s dimensional space of concentrations the state of the system can be represented by the point c = (c1,. .., cs). Evidently this point can only move in an r dimensional subspace if the composition changes are due to only r independent reactions. The orientation of this subspace is given by the matrix of stoicheiometric coefficients, but its position is fixed by the initial composition. Indeed the stoicheiometry of the possible reactions can be determined from the atomic matrix by the following theorem this does not of course imply that all, or even any, of the possible reactions do take place. 

We wish to show next that the stoicheiometry may be deduced from a given set of kinetics to within the usual arbitrary rotation. Whilst the deduction of the stoicheiometry from the atomic matrix can only give the maximum number of reactions, the deduction from a given kinetics may show that not all the possible reactions are taking place. 

This will be called the module of elements. The molecular species (without representation of structure) are defined as linear combinations of these elements (P, Def. 1) and we shall consider mixtures of s distinguishable molecular species. The mixture module consists of all lists of s species,, , j /2, ..., sQ, the only restriction being that they be formed from the fixed set of elements, 33i,..., %T. Let p = [/3(] be a matrix of full rank t (t S s), which may be called the atomic matrix of the mixture /3. We lose no generality by... 

The atomic matrix )8 induces a mapping from the module into the module 9ft by mapping the element (si, eT 2 v) of (5 into the element... 

When the number of independent chemical reactions equals C - p, where p is the rank of the atom matrix (mjk), Gibbs free energy is minimized subject to atom balance constraints ... 

Brinkley (4 postulated C species at equilibrium, p species, referred to as "components," were selected to have linearly independent formula vectors, where p is the rank of the atom matrix, (mjk), and Yj is the formula vector for the jth species, [mj, mj2f...mjE]. Given the choice of components, the stoichiometric coefficients for an independent set of chemical reactions are computed ... 

The formalism of electron-atom scattering has been extensively dealt with elsewhere./27,28,29/ We shall only recall its main features here. Because of the assumed spherical symmetry, the partial-wave scattering approach is convenient. Namely, an incoming spherical wave h[2 kr) y/ (r), (—can scatter only into the outgoing spherical wave hf kr) Y/"(r) (here hf and hf2 are Hankel functions of the first and second kinds, k = 2n/h(2mE) i, E is the kinetic energy and r — r ). This occurs with amplitude t( (f is an element of the diagonal atomic <-matrix), which is related to the phase shifts 6l through... 

The observed energies of the E2 j)cak, as compiled by Van Vechten (1969b), plotted against the prediction based upon atomic matrix elements and atomic term values from the Solid State Table. The empty circles arc the homopolar systems Sn, Gc, Si, and C, in order of increasing gap. 

The hybrids are then written in terms of atomic states, as in Eq. (3-1), and familiar properties of the angular momentum operators are used to evaluate the intra-atomic matrix elements. In terms of our notation in Eq. (3-1), (Py = ih, but... 

Pantelides and Harrison (1975) noted that if one took F, to be identically zero, and wrote both F4 and F5 in terms of atomic matrix elements, was found to be... 

You should notice that there are also nearest-neighbor atomic matrix elements between a bond orbital and the next-nearest-neighbor antibonding orbital however, they are are of order 0, and only contribute to the energy to order 0 thus they do not affect Ci. We have, in fact, included all terms of order 0, so the calculation of Cj is exact. 

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